Regression with n$\to$1 by Expert Knowledge Elicitation

We consider regression under the "extremely small $n$ large $p$" condition. In particular, we focus on problems with so small sample sizes $n$ compared to the dimensionality $p$, even $n\to 1$, that predictors cannot be estimated without prior knowledge. Furthermore, we assume all prior knowledge that can be automatically extracted from databases has already been taken into account. This setup occurs in personalized medicine, for instance, when predicting treatment outcomes for an individual patient based on noisy high-dimensional genomics data. A remaining source of information is expert knowledge which has received relatively little attention in recent years. We formulate the inference problem of asking expert feedback on features on a budget, present experimental results for two setups: "small $n$" and "n=1 with similar data available", and derive conditions under which the elicitation strategy is optimal. Experiments on simulated experts, both on simulated and genomics data, demonstrate that the proposed strategy can drastically improve prediction accuracy.